Say I had an expression that went like this [tex]\frac{ 5 \frac{x}{0} }{3 \frac{x}{0} }[/tex] Can I divide those [itex]\frac{x}{0}[/itex] terms or do they make the expression undefined?
I wouldn't say you can't divide them out so much as what you wrote is just plain wrong, being polite about it - the symbols make no sense as written.
I think he means [tex]\frac{ 5 \cdot \frac{x}{0} }{3 \cdot \frac{x}{0} }[/tex] to be read "5 times x over 0...", not "5 and x zeroths..." - Warren
But is still makes no sense. x/0 is not a well-defined symbol in the real number system that one can manipulate like this.
1/0.1 is tha same as 1*10 1/0.01 is tha same as 1*100 1/0.001 is tha same as 1*1000 and so on ... 1/0 is the same as 1*oo and in both cases we are no longer in a finite system. oo is a general notation for infinity therefore 1/0 is also a general notation for infinity. Please look at: http://mathworld.wolfram.com/Infinity.html
Note for others, in the link given it goes points out: "Informally,[itex]1 / \infty = 0[/tex] , a statement which can be made rigorous using the limit concept" You can't just say: [tex]\frac{1}{\infty} = 0[/tex] or any manipulation of that as and think it is mathematically true.